Now that you’ve found a polynomial to approximate your function, how good is your polynomial?

Find the 6th degree Maclaurin polynomial for x

xxf

sin

For what values of x does this polynomial best follow the curve? Where does the polynomial poorly follow the curve?

What are the limitations of graphically analyzing a Taylor polynomial?

Suppose that a function f(x) has derivatives at x = 0 given by the formula:

n

nn nf

3

!10

Write the first few terms of the Taylor series centered at x = 0 for this function.

Write the 4th degree Taylor polynomial for f centered at x = 0.

Estimate the error in using the 4th degree polynomial to approximate f(0.2).

Error Bounds for ALTERNATING Series

Example Write the 4th degree Maclaurin polynomial for:

xy cos

Show that this polynomial approximates cos(.9) to better than 1 part in 1000.

Example Consider the power series:

...21...642 12753 nn xnxxxxxf

What is the maximum error in truncating the function after the 4th term on the interval -.5 < x < .5?

Example Suppose that f is a function such that f(2)=3 and : nnf 22

Write the 3rd degree Taylor polynomial for f centered at x = 2.

Estimate f(2.1). What is the maximum difference between your estimate and the actual value of f(2.1)?

What is the 4th degree Maclaurin polynomial for ?2xey

Using the polynomial, estimate y(.2). How good is your estimate? Why we can’t we use our usual method to estimate the error?

Taylor’s Theorem

The difference between a function at x and it’s nth degree Taylor polynomial centered a is:

1

1

!1

nn

n axn

cfxR

for some c between x and a.

Taylor’s Theorem is an existence theorem. What does that mean?

What other existence theorems have we seen in Calculus?

Recall our 4th degree polynomial for and our estimate for y(.2).

2xey

Use Taylor’s Theorem to estimate the difference between our estimate and the true value of y(.2).

Lagrange Error Bound

Choose M to be at least as big as the maximum value of the n+1 derivative on the interval x to a.

1

1

n

xMxR

n

Example

Write the 3rd degree Taylor polynomial, P(x), for centered at x= 0.

xey 2

Estimate the error in using P(.2) to approximate . 4.0e